Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.
(sin x + cos x) ^2
a. 1+2sinxcosx
b. sec^2x−tan^2x+2cosxsinx
c. sec x + 2 sin x/sec x
d. sin^2x+cos^2x
e. 1+2cos (pi/2 -x) cosx
take a look at the related questions below. In particular,
http://www.jiskha.com/display.cgi?id=1468255540
ah ok, so would it be answer a?
Read the question carefully. It said
which of the following is not equivalent?
Actually, both c & d are not
To simplify the expression (sin x + cos x) ^2, we can use the distributive property of multiplication.
Start by expanding the expression:
(sin x + cos x) ^2 = (sin x + cos x) * (sin x + cos x)
Using the distributive property, multiply each term in the first set of parentheses by each term in the second set of parentheses:
(sin x + cos x) * (sin x + cos x) = sin x * sin x + sin x * cos x + cos x * sin x + cos x * cos x
Now, simplify each term by applying the fundamental identities:
1. sin x * sin x = sin^2x (using the identity sin^2θ = (1 - cos(2θ))/2)
2. sin x * cos x = 1/2 * sin(2x) (using the identity sin(2θ) = 2sinθcosθ)
3. cos x * sin x = 1/2 * sin(2x) (using the same identity as above)
4. cos x * cos x = cos^2x (using the identity cos^2θ = (1 + cos(2θ))/2)
After simplifying each term, we have:
sin^2x + 1/2 * sin(2x) + 1/2 * sin(2x) + cos^2x
Combine like terms:
sin^2x + 1 * sin(2x) + cos^2x = sin^2x + sin(2x) + cos^2x
Now, we can use the Pythagorean identity sin^2θ + cos^2θ = 1:
sin^2x + sin(2x) + cos^2x = 1 + sin(2x)
So, the simplified expression is 1 + sin(2x).
Now, let's determine which of the following options is not equivalent:
a. 1 + 2sin(x)cos(x) (This is equivalent, as 2sin(x)cos(x) can be further simplified to sin(2x))
b. sec^2(x) − tan^2(x) + 2cos(x)sin(x) (This is equivalent, as sec^2(x) - tan^2(x) can be simplified to 1)
c. sec(x) + 2sin(x)/sec(x) (This is equivalent, as 2sin(x)/sec(x) can be simplified to 2cos(x))
d. sin^2(x) + cos^2(x) (This is the original expression, so it is equivalent)
e. 1 + 2cos(pi/2 - x)cos(x) (This is not equivalent, as 2cos(pi/2 - x)cos(x) cannot be further simplified)
Therefore, option e. 1 + 2cos(pi/2 - x)cos(x) is not equivalent.